I have a very simple HTML table with 4 columns:
Facility Name, Phone #, City, Specialty
I want the user to be able to sort by Faci
My answer would be "be careful". A lot of jQuery table-sorting add-ons only sort what you pass to the browser. In many cases, you have to keep in mind that tables are dynamic sets of data, and could potentially contain zillions of lines of data.
You do mention that you only have 4 columns, but much more importantly, you don't mention how many rows we're talking about here.
If you pass 5000 lines to the browser from the database, knowing that the actual database-table contains 100,000 rows, my question is: what's the point in making the table sortable? In order to do a proper sort, you'd have to send the query back to the database, and let the database (a tool actually designed to sort data) do the sorting for you.
In direct answer to your question though, the best sorting add-on I've come across is Ingrid. There are many reasons that I don't like this add-on ("unnecessary bells and whistles..." as you call it), but one of it's best features in terms of sort, is that it uses ajax, and doesn't assume that you've already passed it all the data before it does its sort.
I recognise that this answer is probably overkill (and over 2 years late) for your requirements, but I do get annoyed when developers in my field overlook this point. So I hope someone else picks up on it.
I feel better now.